PARAMETRIC SAR IMAGE FORMATION - A PROMISING APPROACH TO RESOLUTION-UNLIMITED IMAGING. Yesheng Gao, Kaizhi Wang, Xingzhao Liu

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1 20th European Signal Proessing Conferene EUSIPCO 2012) Buharest, Romania, August 27-31, 2012 PARAMETRIC SAR IMAGE FORMATION - A PROMISING APPROACH TO RESOLUTION-UNLIMITED IMAGING Yesheng Gao, Kaizhi Wang, Xingzhao Liu Shanghai Jiao Tong University Department of Eletroni Engineering ysgao@sjtu.edu.n ABSTRACT A parametri SAR image formation sheme is proposed in this paper. Based on a model desribing the reeived signal, the parameters of the model an be used to haraterize the illuminated sene. A realization of parameter estimation via atomi deomposition is presented. The estimated parameters are then mapped onto the iso-range-doppler grid to form a SAR image. Simulation results demonstrate that this sheme an provide an approah to resolution-unlimited imaging. Index Terms Syntheti aperture radar SAR), image formation, parameter estimation, atomi deomposition 1. INTRODUCTION Syntheti aperture radar SAR) uses the relative motion between an antenna and the illuminated sene to form high resolution images. The antenna is usually mounted on a moving platform, inluding airraft and spaeraft [1]. Conventional SAR imaging algorithms have been developed based on the data olletion geometries. The range resolution is onstrained by the transmitted signal bandwidth, while the azimuth resolution is determined by the syntheti aperture length or the syntheti angle. SAR image formation an also be viewed as a parameter estimation problem [2]. DeGraaf disussed SAR image formation using modern 2-D spetral estimation methods, and gave a omprehensive omparison of these methods [2]. Gerry proposed a parametri model for radar sattering in two dimensions, and presented an approximate maximum likelihood ML) algorithm for estimating the parameters [3]. Li gave a model of the signal refleted from dihedrals and trihedrals, and implemented image formation and target feature extration using spetral estimation methods [4, 5]. We present a parametri SAR image formation sheme in this paper. Using a point satterer assumption, a parametri model is used to desribe the SAR signal. Eah satterer is haraterized by a set of parameters, and the parameters are estimated by signal proessing tools. We take atomi deomposition as an example for parameter estimation to demonstrate the feasibility of the proposed sheme. The remainder of this paper is organized as follows. In setion II, the fundamentals of SAR imaging are reviewed. In setion III, the model of the SAR signal is introdued. In setion IV, a realization of the proposed parametri SAR imaging sheme is disussed. Atomi deomposition is used to estimate parameters, whih is mapped to form a SAR image. In setion V, simulation results are given. A summary is onluded in setion VI. 2. FUNDAMENTALS OF SAR IMAGING flight path Fig. 1. Geometry of SAR Data olletion. The data olletion geometry is illustrated in Fig. 1. The plane onstituted by the flight path and the sene enter is labeled as an x-y Cartesian oordinate system. The origin of the oordinate system, O, is at the sene enter. Assume that t = 0 is the time of losest approah, azimuth time from t begin to t end is a oherent proessing aperture, t denotes the azimuth time evaluated at the aperture enter. The instantaneous slant range Rt) an be expressed as Rt) = R 2 v s t ) 2 + v s t) 2 1) EURASIP, ISSN

2 where v s is the platform veloity, R is the distane between the antenna phase enter APC) and the target at t = t. The spatial resolution in the range dimension is ρ r = K r 2) 2B where is light veloity, B is the bandwidth of the transmitted signal. K r is a mainlobe broadening fator introdued by the aperture weighting funtion used for sidelobe ontrol and by any low-frequeny phase errors present in the signal. The expression for azimuth resolution is λ ρ a = K a 3) 2θ syn where λ is the wavelength, θ syn is the angle during whih the target is illuminated in the oherent proessing aperture. K a is a mainlobe broadening fator introdued by aperture weighting for sidelobe ontrol of the system impulse response and by residual amplitude and phase errors. The area determined by ρ r and ρ a is defined as resolution ell. SAR reeives the signal refleted from many resolution ells simultaneously, but is able to separate the returns from eah resolution ell by using image formation algorithms. Conventional algorithms have been developed based on data olletion geometries, and Fourier transform provides a hane of their effetive implementations. Suh algorithms inlude range-doppler algorithm RDA), hirp saling algorithm CSA), polar format algorithm PFA) and ω-k algorithm ωka). 3. MODEL OF THE SAR SIGNAL Unlike onventional image formation algorithms, we desire to present a parametri image formation sheme. In this setion, the model of the SAR signal is desribed. We onentrate on azimuth signal fousing, the method of parameter estimation disussed in the next setion an be extended to range dimension straightforwardly. After demodulation, the signal refleted from a point satterer an be written as sτ, t) =Aω r τ 2Rt) ) ω a t t ) 4) e jπk rτ 2Rt) ) 2 e j 4π λ Rt) where τ and t are fast and slow time, ω r ) and ω a ) are the range and azimuth envelope, A is a onstant, k r is the hirp rate of transmitted signal. The range-ompressed signal is st) = Ap r τ 2Rt) ) ω a t t )e j 4π λ Rt) 5) where p r ) is the range-ompressed envelope. Rt) is alulated using equation 1). Aording to the atual physial proedure of SAR imaging, a mathematial funtion an be used to fit Rt), and then a model of the signal an be onstruted. Rt) is approximated by R p t) = R + a 1 t t ) + a 2 t t ) ) The model of the SAR signal an be expressed as s p t; θ p ) = Ap r τ 2R pt) )ω a t t )e j 4π λ R pt) where θ p = [T, t, a 1, a 2, ], T denotes the temporal support of ω a ). The signal an be foused in azimuth by estimating θ p. The order of the polynomial in 7) is determined by the relationship between the resolution and the geometri error. That is, it is determined by whether or not the differene between the atual and the presumed geometry is omparable to the spatial resolution. Generally, the finer the spatial resolution is, the higher the order of the polynomial needed. The signal model of a group of point satterers an be written as s Σ t; θ i p) = i = i s i pt) A i p i r τ 2Ri pt) ) 7) ω i at t i )e j 4π λ Ri p t). These satterers an be reonstruted one by one in a SAR image. 4. ATOMIC DECOMPOSITION-BASED IMAGE FORMATION ALGORITHM 4.1. Parameter estimation In this setion, we give a realization of the parametri image formation sheme. The parameters are estimated via atomi deomposition AD) [6, 7]. A ditionary is onstruted to implement the AD. Eah atom in the ditionary is unity energy and has the form as below. at; θ p ) = 1 ret t t T T )e j 4π 8) λ Rpt). 9) To estimate θ p, a ost funtion named mathing degree indiator MDI) is defined as Mθ p ) = t +T/2 t T/2 s t, Rθ p )) a t; θ p ) dt 2 10) 2517

3 where s t, Rθ p )) is the range-ompressed signal along RMC Rθ p ), and s t, Rθ p )) is the form with its energy normalized to unity. Eah θ p an be used to alulate a range migration urve RMC). The parameter estimation an be regarded as a proedure of finding the RMC along whih the signal an math the model best. That is, it is to find the θ p maximizing equation 9). The parameter estimation is demonstrated in Fig. 2. The best-mathed parameter vetor θ p an be obtained by θ p = arg max Mθ p ). 11) The intensity of the satterer present in the final SAR image is where Q = t+t/2 t T/2 A in = Q Mθ p) 12) s t, Rθ p) ) s t, Rθ p) ) 2 dt. 13) The best-mathed parameter vetor θ p is estimated as follows. Step 1: Construt the ditionary of atoms with parameter vetors {θ p }. Step 2: Calulate the range migration Rθ p ) of eah θ p in the ditionary aording to equation 6). Step 3: Searh over the data matrix for the best-mathed parameter vetor θ p using equation 11). Step 4: The intensity of the satterer is obtained using equation 12). The parameter estimation an be ontinued until the residual energy is over E th or the ahieved MDI is below M th. We assume that the obtained θ p in eah iteration orresponds to a point satterer, whih is reonstruted in the final SAR image. iso-doppler line nadir line iso-range line Fig. 3. Grid formed by iso-range and iso-doppler lines. iso-doppler ontours in the ground plane [1]. For a SAR radar flying along an ideal trajetory, a onstant-range sphere is the sphere entered at the APC. Constant-range spheres interset the flat ground plane as irles entered at the APC nadir point, marked as the solid blak irle in Fig. 3. Eah of these irles is alled iso-range line, as the dashed lines in Fig. 3. The satterers plaed on the same iso-range line have the same distane to the APC. A onstant Doppler surfae is the onial surfae with apex at the APC. Constant-Doppler ones interset the flat ground plane as hyperbolas. Eah of these hyperbolas is alled iso-doppler line, as the solid lines in Fig. 3. The satterers plaed on the same iso-doppler line have the same Doppler entroid. The image grid is formed by these two sets of lines and is alled iso-range-doppler grid. In the AD-based image formation algorithm, a SAR image is formed by mapping the estimated parameters onto the grid. The azimuth position of a satterer is determined by its Doppler entroid, i.e., f = 2 λ a 1. The intensity of the satterer A in is obtained in Step 4 of the parameter estimation. 5. SIMULATION RESULTS 5.1. Image formation of a single satterer Fig. 2. Proedure of parameter estimation parameter mapping SAR image an be viewed as a projetion of the sene radar ross setion RCS) onto the grid formed by iso-range and The primary parameters are listed in Table 1. The point satterer is plaed at the enter of the illuminated sene. In this simulation, we adopted quadrati phase in equation 9). Fig. 4 is the imaging result with E th = 2% of the original energy. Fig. 5 is the imaging result with E th = 5% of the original energy. In both simulations, M th is set at 0.8. The graphi proessing unit GPU) is used in the alulation owing to its powerful parallel omputation apability [8]. In Fig. 5, the point satterer is haraterized by a single parameter vetor, thus it is reonstruted as a pixel in the final image. When E th dereased to 2% of the original energy, 2518

4 Table 1. Simulation parameters Parameter Value Wavelength m Bandwidth 800 MHz Pulse length 10 µs PRF 1000 Hz Platform veloity 100 m/s 5.2. Image formation of multi-targets azimuth range Azimuth Fig. 6. Image formed using the CSA. Range Fig. 4. Imaging result E th = 2% of the original energy). more parameter vetors an be estimated, the orresponding intensity of the seond parameter vetor is -35 db below the first one. This results from the estimation error by using the signal model. Unlike the onventional SAR images, the parametri image formation sheme an generate resolutionunlimited images. Stritly speaking, resolution-unlimited imaging an be onsidered the SAR image is formed by reonstruting satterers) by satterers). The resultant image is suitable for target detetion and feature extration, although it may not be visually satisfatory. Azimuth Range Fig. 5. Imaging result E th = 5% of the original energy). azimuth range Fig. 7. Image formed using AD-based algorithm. In this simulation, seven point targets are plaed into the letter T, and the spae of two adjaent point targets is twie the range resolution. The key parameters of radar system are listed in Table 1, quadrati phase is also adopted. In this simulation, E th is set at 5% of the original energy, and M th is set at 0.8. Fig. 6 shows an image made using the hirp saling algorithm [9]. An image made using the AD-based imaging algorithm is shown in Fig. 7. The image quality of Fig. 6 is deteriorated due to the mainlobe and sidelobe effets, whih is an intrinsi drawbak of Fourier SAR imaging. In Fig. 7, over 95% of a target s energy is onentrated on a pixel of the image. 6. CONCLUSION We present a parametri SAR image formation sheme in this paper and give its realization via atomi deomposition. With the assumption of point satterer, eah set of model parameters is estimated to haraterize a single satterer. From the simulation results, a point satterer an be presented as a single pixel in the final image. That is, the resolution is 2519

5 not onstrained by the radar system and the syntheti aperture length as the onventional algorithms do. However, the AD-based algorithm is more omputationally omplex than the onventional ones. Our further work will onentrate on algorithm struture and performane optimization. 7. REFERENCES [1] Walter G. Carrara, Ronald M. Majewski, and Ron S. Goodman, Spotlight Syntheti Aperture Radar: Signal Proessing Algorithms, Arteh House, July [2] S.R. DeGraaf, SAR imaging via modern 2-D spetral estimation methods, IEEE Transations on Image Proessing, vol. 7, no. 5, pp , May [3] M.J. Gerry, L.C. Potter, I.J. Gupta, and A. Van Der Merwe, A parametri model for syntheti aperture radar measurements, IEEE Transations on Antennas and Propagation, vol. 47, no. 7, pp , Jul [4] Renbiao Wu, Jian Li, Zhaoqiang Bi, and P. Stoia, SAR image formation via semiparametri spetral estimation, IEEE Transations on Aerospae and Eletroni Systems, vol. 35, no. 4, pp , Ot [5] Zhaoqiang Bi, Jian Li, and Zheng-She Liu, Super resolution SAR imaging via parametri spetral estimation methods, IEEE Transations on Aerospae and Eletroni Systems, vol. 35, no. 1, pp , Jan [6] O.A. Yeste-Ojeda, J. Grajal, and G. Lopez-Risueno, Atomi deomposition for radar appliations, IEEE Transations on Aerospae and Eletroni Systems, vol. 44, no. 1, pp , Jan [7] A. Bultan, A four-parameter atomi deomposition of hirplets, IEEE Transations on Signal Proessing, vol. 47, no. 3, pp , Mar [8] J.D. Owens, M. Houston, D. Luebke, S. Green, J.E. Stone, and J.C. Phillips, GPU omputing, Proeedings of the IEEE, vol. 96, no. 5, pp , May [9] R.K. Raney, H. Runge, R. Bamler, I.G. Cumming, and F.H. Wong, Preision SAR proessing using hirp saling, IEEE Transations on Geosiene and Remote Sensing, vol. 32, no. 4, pp , Jul